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Turbulence Closure in RANS and Flow Inference around a Cylinder using PINNs and Sparse Experimental Data

Z. Zhang, K. Shukla, Z. Wang, A. Morales, T. Käufer, S. Salauddin, N. Walters, D. Barrett, K. Ahmed, M. S. Triantafyllou, G. E. Karniadakis

Abstract

Traditional Reynolds-averaged Navier-Stokes (RANS) closures, based on the Boussinesq eddy viscosity hypothesis and calibrated on canonical flows, often yield inaccurate predictions of both mean flow and turbulence statistics. Here, we consider flow past a circular cylinder over a range of Reynolds numbers (3,900-100,000) and Mach numbers (0-0.3), encompassing incompressible and weakly compressible regimes, with the goal of improving predictions of mean velocity and Reynolds stresses. To this end, we assemble a cross-validated dataset comprising hydrodynamic particle image velocimetry (PIV) in a towing tank, aerodynamic PIV in a wind tunnel, and high-fidelity spectral element DNS and LES. Analysis of these data reveals a universal distribution of Reynolds stresses across the parameter space, which provides the foundation for a data-driven closure. We employ physics-informed neural networks (PINNs), trained with the unclosed RANS equations, to infer the velocity field and Reynolds-stress forcing from boundary information alone. The resulting closure, embedded in a forward PINN solver, significantly improves RANS predictions of both mean flow and turbulence statistics relative to conventional models.

Turbulence Closure in RANS and Flow Inference around a Cylinder using PINNs and Sparse Experimental Data

Abstract

Traditional Reynolds-averaged Navier-Stokes (RANS) closures, based on the Boussinesq eddy viscosity hypothesis and calibrated on canonical flows, often yield inaccurate predictions of both mean flow and turbulence statistics. Here, we consider flow past a circular cylinder over a range of Reynolds numbers (3,900-100,000) and Mach numbers (0-0.3), encompassing incompressible and weakly compressible regimes, with the goal of improving predictions of mean velocity and Reynolds stresses. To this end, we assemble a cross-validated dataset comprising hydrodynamic particle image velocimetry (PIV) in a towing tank, aerodynamic PIV in a wind tunnel, and high-fidelity spectral element DNS and LES. Analysis of these data reveals a universal distribution of Reynolds stresses across the parameter space, which provides the foundation for a data-driven closure. We employ physics-informed neural networks (PINNs), trained with the unclosed RANS equations, to infer the velocity field and Reynolds-stress forcing from boundary information alone. The resulting closure, embedded in a forward PINN solver, significantly improves RANS predictions of both mean flow and turbulence statistics relative to conventional models.

Paper Structure

This paper contains 26 sections, 21 equations, 29 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Data Generation and Cross Validation
  4. Hydrodynamic Particle Image Velocimetry
  5. Aerodynamic Particle Image Velocimetry
  6. Numerical Simulation
  7. Cross Validation
  8. Physics-Informed Data Postprocessing
  9. Helmholtz Decomposition and Divergence Correction
  10. Pressure Computation from $\mathbf{U}$ and $\mathbf{F}_{\mathrm{m}}$
  11. Reconstruction of the Corrected Forcing $\mathbf{F}_{\mathrm{new}}$
  12. Flow Inference
  13. Flow inference with unclosed RANS equation
  14. Flow inference with Helmholtz and Turbulence augmented approach using boundary data for Re=3900
  15. PINNs architecture, hyper-parameters and results on flow inference
  16. ...and 11 more sections

Figures (29)

  • Figure 1: Overview of the paper. (1) PIV and DNS/LES are used to establish a dataset of flow past a cylinder. The range of the key parameters, $Re$ and $Ma$, is listed. (2) PINNs are used to infer the flow fields within a domain $\varOmega$ based on the unclosed RANS equation and the boundary conditions at $\partial \varOmega$ for both incompressible and weakly compressible regimes. (3) Data-driven turbulence closure model is built and integrated with the forward PINN solver, investigating the accuracy of both velocity and Reynolds forcing fields.
  • Figure 2: Hydrodynamics PIV: Overview of the mean velocity components $U$ and $V$, and the Reynolds stress components $\overline{u'u'}$, $\overline{u'v'}$, $\overline{v'v'}$ for $Re = 10,000$, $Re = 20,000$, $Re = 30,000$ (left to right).The white region was masked during the processing. The black region depicts the cylinder.
  • Figure 3: Aerodynamic PIV: Time-averaged velocity components and Reynolds stresses measured from Re = 6,500 - 100,000. White regions indicate areas where vector data could not be resolved, and the black portion marks the location of the cylinder.
  • Figure 4: Comparison between time-averaged DNS and hydrodynamic PIV fields. The mean velocity $U$ and $V$ and the Reynolds stresses $\overline{u'u'}$, $\overline{u'v'}$, and $\overline{v'v'}$ are shown. The Reynolds number for DNS is $Re_{DNS}=11\,000$, while the Reynolds number for PIV is $Re_{PIV}=10\,000$.
  • Figure 5: Comparison between time-averaged DNS and hydrodynamic PIV results at four streamwise positions $x=1,2,3,4$. The mean velocity $U$ and $V$ and the Reynolds stresses $\overline{u'u'}$, $\overline{u'v'}$, and $\overline{v'v'}$ are shown. The Reynolds number for DNS is $Re_{DNS}=11\,000$, while the Reynolds number for PIV is $Re_{PIV}=10\,000$.
  • ...and 24 more figures